Question 393: Centre of Mass and Rotational Motion

Question

$\mathrm{A}, \mathrm{B}$ and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ axis is $\frac{x}{15} \mathrm{I}$, where I is the moment of inertia of the disc about its diameter. The value of $x$ is ____________.

Accepted Answer

All bodies have same mass and same radius.
A $ightarrow$ Disc
B $ightarrow$ Solid sphere
$\mathrm{C} ightarrow$ Spherical shell
$$\begin{aligned}
& 	ext { and, } \mathrm{I}=\frac{\mathrm{MR}^2}{4} \
\mathrm{I}_{\mathrm{PQ}} & =\frac{\mathrm{MR}^2}{4}+\left(\frac{2}{5} \mathrm{MR}^2+\mathrm{MR}^2ight)+\left(\frac{2}{3} \mathrm{MR}^2+\mathrm{MR}^2ight) \
\mathrm{I}_{\mathrm{PQ}} & =\frac{15 \mathrm{MR}^2+24 \mathrm{MR}^2+60 \mathrm{MR}^2+40 \mathrm{MR}^2+60 \mathrm{MR}^2}{60} \
\mathrm{I}_{\mathrm{PQ}} & =\frac{199}{60} \mathrm{MR}^2=\frac{199}{15}\left(\frac{\mathrm{MR}^2}{4}ight) \
& =\frac{199}{15} \mathrm{I}
\end{aligned}$$

Centre of Mass and Rotational Motion

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